- Heat oil in a heavy large skillet over medium-high heat until it shimmers. Cook onions with ½ teaspoon salt and ¼ teaspoon fresh ground black pepper covered, stirring occasionally until light gold in color, approximately 5 to 10 minutes.
- Reduce heat to medium and continue cooking until onions are soft and a deep golden brown, stirring often, approximately 10 to 15 minutes longer. Season to taste with salt and pepper. Transfer to a small bowl and keep warm, covered.
- Chop bacon into ½ to 1 inch pieces. In a large bowl place burger meat on bottom of bowl and top with chopped bacon. Sprinkle with Worcestershire sauce. Combine gently (do not over mix) into 4 patties approximately 1 inch thick.
- Brush faces of rolls with melted butter.
- Oil grill rack and sear burgers for 2 to 3 minutes per side, then grill burgers (on grill but away from direct heat) to desired temperature.
- Top with cheese, cover and allow to melt. Transfer burgers to a clean platter and cover, allowing to rest.
- Grill buns, buttered side down until grill marks appear.
- Add caramelized onions on top cheese. Finish with tomato and arugula.